WCPM Seminar Series, Feb. 12, 2015 - Warwick
Electronic, thermal, and
thermoelectric transport in
nanostructures
Neophytos Neophytou
School of Engineering, University of Warwick, Coventry, U.K.
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics: control scattering
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
2
Why nano ?
New low-dimensional materials:
 2D ultra thin layers
 1D nanowires
 0D quantum dots
 2D graphene, 1D carbon nanotubes
Design degrees of freedom for design:
 Length scale - geometry
 Quantum effects (electrons behave differently)
 Atomistic effects
thin layers
Uchida et al., IEDM 03
nanowires
Trivedi, 2011
quantum dots
2D graphene
nanotube
3
Applications for nanodevices
Breus et al., Nano Lett., 2009
Ahn et al., Nano Lett., 2010
Robert Chau, Intel, 2005
Bio-sensors
Nano-transistors
Bio-illumination /
drug delivery
Kim et al., Nano Lett., 2011
Lu et al.,
Nano Lett., 2009
Optoelectronics
Boukai et al., Nature 2008
Photovoltaics
Thermoelectrics
4
Nano-design for transistors
Dr
ai
n
gate
source
Channel
drain
Length scale (the shorter, the faster)
60mV/dec
e
e
So
ur
c

Gate
Lch
LG
HSi
WSi
3D geometry (better electrostatics)
<40mV/dec
[100]
[110]
[111]
experiments
Appenzeller, PRL, 2004 (IBM)
Quantum effects (band to band tunneling)
Neophytou, Nano Lett., 10, 4913, 2010, APL 2011
Atomistic effects (bandstructure)
5
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
6
Attempt similar design for thermoelectrics
e
e
HOT
Electrical
conductivity
heat
COLD
Seebeck
coefficient
 S 2T
ZT 
e  l
Snyder et al., Science, 2008, Nat. Mat., 2008.
Electronic
thermal
conductivity
Lattice
thermal
conductivity
 15 TWatts of heat are lost, but
 State of the art: ZT~1.5 (need ZT~4)
 Rare earth, toxic, expensive materials
7
Abundance issues with good TE materials
http://pubs.usgs.gov/fs/2002/fs087-02/
Abundance issues for Te, toxicity for Pb
8
Design targets for nano-TE materials
Sharp peaks in DOS(E)
S~
d
DOS( E )
dE
Hicks and Dresselhaus -1993, Dresselhaus - 2001
 Nanostructuring phonon engineering
 Scatter phonons only
 Low dimensionality –
improves S
ZT 
S 2T
k e  kl
9
How to proceed further ?
Case for Si:
Bulk : 140 W/mK, ZT=0.01
NWs: 1-2 W/mK, ZT~1
Vineis et al., Adv. Mater. 22, 3790, 2010
- κl reduction benefits are reaching their limits (easily)
- we need to look into σS2
10
This talk’s focus
(1)
(2)
Electronic properties: model and simulations
Interplay between σ, S at the nanoscale
(Si @ T=300K)
Phonon properties: model and simulations
Possibility of further reduction in κl
(100) , (110), (111)
(100), (110), (112)
<100> , <110>
Source
[100]
[110]
[111]
nanowires
Channel
Drain
[100]
[110]
[111]
thin layers
nanocrystalline
nanomeshes
graphene
11
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
12
Channel description: Atomistic Tight-Binding
[100]
[110]
[111]
NN sp3d5s*-SO
Based on Localized Atomic Orbitals
Suitable for:
 Structure deformations, strain
 Material variations, heterostructures
The bulk bandstructure
 Needs a set of fitting parameters
 Computationally expensive
 Compromise: between ab-initio methods and continuum methods
 Able to handle 10s of thousands of atoms
13
Valleys-from bulk, to quantum wells, and to NWs
<001>
two-fold
perpendicular valleys
four-fold
in-plane valleys
<010>
Si
3D
<100>
Γ
Γ
(001)
2D
(001)
1D
Projection to 1D
4 X equivalent
valleys at Γ
14
Electronic structure examples
(100) , (110), (111)
(100), (110), (112)
<100> , <110>
Source
Channel
Drain
[100]
[110]
[111]
[100]
[110]
[111]
NN sp3d5s*-SO
Conduction band
Valence band
1st Brillouin Zone
15
Length scale dependent properties
mass variation
D=3nm
D=12nm
[100]
[111]
 Electronic structure is geometry dependent
NN, IEEE TED, 2008, Nano Lett., 2010
16
TB coupled to Linearized Boltzmann transport
      vk vk k       k  
k
 g   v     

2



R ( )


  
 f 
 q0 2  d    0      

  
 k BT 
E0


 R
At all κ-point, subbands: vn  E   1 En
k x
 velocity
 density of states
g1nD  E  
1
2
1
vn  E 
 0
k B R1
S
q0 R 0
2
1





2
R
k T
 e  B 2  R 2    0 

q0 
R


17
Linearized Boltzmann transport: 2D
(100) , (110), (111)
(100), (110), (112)
<100> , <110>
Source
Channel
Drain
[100]
[110]
[111]
 Relaxation times
(of every k-state, at every subband)
 phonon scattering (acoustic/optical)
 surface roughness scattering
 ionized impurity scattering
18
Basic features for TE coefficients – simple guidelines

 f 0 
  q0  d   
  

  
E0
2
exp.
σ ~ 1/meff * exp(-ηF)

   EF 
 f 0 
S
d  
   



 E0   
k
T
 B 
kB q0
linear
F  E0  EF
S ~ ηF
Power factor maximum around Ef
19
Design direction for σS2 at low dimensions
well
barrier
well
barrier
Change geometry at the same charge density:
in-plane SLs
cross-plane SLs
core-shell NWs
SL NWs
σ ~ 1/meff * exp(-ηF)
From:
3D to 1D
nano-dots
[100]
in lattices
Nanograins
[110]
S ~ ηF
NW lattices/
[111]
anti-lattices (empty)
σS2
ηF =EC-EF
Neophytou and Kosina, PRB, 83, 245305, 2011
20
A thorough investigation for Si: σ determines σS2
σ ~ 1/meff * exp(-ηF)
S ~ ηF
meff
(100), (110), (112)
[100]
[110]
[111]
σS2
S
σ
S
σ
Neophytou and Kosina, PRB, 83, 245305, 2011
Neophytou and Kosina, J. Appl. Phys., 112, 024305, 2012
S
σ
21
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics: control scattering
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
22
Transport: Impurity dominated
k l  2 Wm -1 K -1
~2-4X
reduction
gate all around
(modulation doping)
Modulation doping?
Surface charge transfer?
Gating?
23
Self-consistent computational model
Bandstructure
(states)
(1)
E(k)
Equilibrium statistics
(state filling - charge)
(2)
EF
Uscf
oxide
C
H
A
N
N
E
L
Poisson
Charge
Potential
gate
Boltzmann
transport


(3)
(4)
24
Hole dispersions under confinement
p-type
[110] NW
D=12nm
Low VG
High VG
25
The effect of gating on NW mobility
 Benefit from not using dopants
 Gating seems beneficial, even with surface roughness
(accumulation is achieved with weaker fields)
26
Power factor improvement
Power factor:
 Improved by ~5x
27
Power factor improvement versus diameter
Power factor improvements:
 Still observed at D=20nm we were able to simulate
 Might be retained up to D~40nm
28
Power factor - anisotropy
Strong anisotropy:
 [111] NWs ~2x higher performance than [110]
~3x higher performance than [100]
29
Summary: Design strategies for low-D
1) Optimize the materials bandstructure:
 Best choice of geometry/confinement, ηF
 But in general, use of strain, alloying etc.
2) Avoid the most degrading scattering mechanisms:
 Remove dopant impurities by gate field
 But also modulation doping, would work
30
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
31
Modified Valence Force Field Method (MVFF)
Keating
3 r  d
U  
8
dij2
2
ij
ij
bs
U
jik
bb

2 2
ij
3   
 
jik
8
dij dik
bond-stretching
Modified
U
jik
bs bs
U
2
2
2
2
2
3  rij  dij  rik  dik 
 
8
dij dik
jik
bs bb
bond-bending
U
jikl
bb bb
2
2
3  rij  dij    jik 
 
8
dij dik
3   jik   ikl 
 
8
dij dik2 d kl
cross bond
stretching
cross bond
stretching/
bending
coplanar bond
bending
32
Modified Valence Force Field Method (MVFF)
Keating
Modified
j k
j  k l


1
ij
jik
jik
jik
U     U bs   U bb  U bs bs  U bs bb    U bbjiklbb 
2 iN A  jnni
j , knni
j , k ,lCOPi

2 ij

U
ij
Dmn
 i mnj
rm rn
 Dxxij

Dij   Dyxij
 Dzxij

Dxyij
Dyyij
Dzyij
Dxzij 

Dyzij 
Dzzij 


D   Dl exp iq.Rl   2  q  I  0
l
33
MVFF: Benchmarked to bulk Si
Keating
Modified
34
MVFF: Low-dimensional phonon spectrum
optical
quasiacoustic
acoustic
1D nanowire
2D ultra-thin layer
35
Phonon thermal conductivity (diffusive)
BTE for phonons (bulk formalism)
Umklapp scattering
1
U
 Bi (q) 2 T exp(
C
)
T
Boundary scattering
1
1  p ( q ) v g ,i ( q )

 B ,i ( q ) 1  p ( q ) W
2
p(q)  exp(4q 2  rms
)
P: specularity parameter
P=1, fully specular
P=0, fully diffusive
Higher order scattering
1
 A0T 2
U 2
36
ZT figure of merit
n-type NWs:
ZT~1.4 can be reached
(around 6-7nm)
p-type NWs:
ZT~1.4 can be reached
(around 3nm)
ZT is improved,
but still low !
(Bulk Si ZT is ~0.01)
Just low-D is not enough 37
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
38
Nanocomposite channels for increased Seebeck
Make S and σ really independent?
How to increase both simultaneously?
barrier
well
ηF
EF
LW
superlattices
Si/SiGe
LB
Barriers:
S ~ ηF
σ~exp(-ηF)
Wells:
S ~ EF
σ~exp(-EF)
39
Nanocrystalline Si
Make S and σ really independent?
How to increase both simultaneously?
EF
LG
30nm
 Heavily boron doped
 Annealing steps
 Boron precipitates
(non-uniform distribution)
 charge decreases
 Vb increases
 WD increases
Vb
LGB
EF
LG
Neophytou et al., J. Electr. Mat., 2012, ICT 2012
Collaboration: Prof. X. Zianni (Chalkida) and Prof. D. Narducci (Milano)
LGB
40
Nanocrystalline Si
Make S and σ really independent?
How to increase both simultaneously?
EF
LG
30nm
Vb
LGB
EF
3X
LG
Neophytou et al., J. Electr. Mat., 2012, ICT 2012, nanotechnology, 2013
Collaboration: Prof. X. Zianni (Chalkida) and Prof. D. Narducci (Milano)
LGB
41
Nanocrystalline Si: Simulations vs. experiments
Final
Final
Final
Initial
Initial
Initial
Vbeff=0.06eV
EF
Vb
Initial
LG
LGB
EF
Final
LG
Vbeff=0.09eV
LGB
Simultaneous enhancement in σ and S
Neophytou et al., J. Electr. Mat., 2012, ICT 2012, nanotechnology, 2013
Collaboration: Prof. X. Zianni (Chalkida) and Prof. D. Narducci (Milano)
42
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
43
Si nanomeshes
Nanoporous membranes of single-crystalline Si (“holey” Si)
S T
ZT 
e  l
2
ZT ~ 0.4
Tang et al, Nano Lett. 2010
44
Method: Solve BTE using Monte Carlo
Boltzmann Transport Equation for phonon
Relaxation time approximation
Solve BTE using Monte-Carlo (MC)
boundary scatt.
thermalization
initialization
3-phonon scatt.
Wolf, Neophytou, and Kosina, JAP, 2014
45
Thermal conductivity vs porosity/roughness
random pore arrangement
[4] Randomized pores
[2,3] Ordered arrays (rectangular/hexagonal)
Phonon coherent effects are present !
46
Non-Equilibrium Green’s Function (NEGF)
ELECTROSTATICS
Σ1
Σ2
H+U+ΣS
given n  Uscf
Poisson
Iterate until
convergence
given Uscf  n
- Device Green’s function:
G( E )  [( E  i0 ) I  H     2 ]

1
TRANSPORT
(NEGF)
- Density of states:
1
Trace(GG  ),
2
where   i( -   )
 Very powerful approach
D( E ) 
 Can include scattering (decoherence)
 Can be computationally very expensive
- Transmission:
T ( E )  Trace(  G 2G )

 For both electrons (Hamiltonian) and
phonons (dynamic matrix)
47
Graphene nano-ribbon thermoelectrics
ZT=0
S~
d
DOS( E )
dE
(1)
(2)
(3)
Karamitaheri, Neophytou, Kosina, et al.,
JAP 111, 054501, 2012.
48
Outline
 Introduction – design at the nanoscale
 Nanoscale thermoelectrics – design targets
 Low-dimensional TEs (atomistic tight-binding + BTE)
 Gated thermoelectrics
 Phonons transport for low-D (Modified Valence Force Field)
 ZT figure of merit for low-D channel
 Nanostructured thermoelectrics
 Other studies: Nanomeshes (MC), Graphene (NEGF)
 Future directions and conclusions
49
Hierarchy in geometry
Hierarchical scattering of phonons
Biswas et al. (Kanatzidis group)
Very low κl
barrier
well
Neophytou, Narducci, et al.,
Nanotechnology 2013,
J. Electronic Materials 2014
Very high PF:
2-phase materials: 15 mW/K2m-1
3-phase materials: 22 mW/K2m-1
(~7x compared to bulk)
larger S with 3rd phase
50
ZT figure of merit
 S 2T
ZT 
l
5X
15X
κl=140 W/mK (bulk)
κl=8 W/mK (our nano-grains, calculations)
Put all together:
 ZT~4 ?
placement of dopants
bandstructure
engineering
Reduce κl
κl=1-2 W/mK (as in NWs)
51
Transport in multi-phase materials using NEGF
 We start with 1D
 Then extend to 2D
Transport through wells and barriers:
 Include acoustic and optical phonons
 Energy relaxation as current flows
 Include quantum effects.
 Can retrieve ballistic and diffusive regimes
Red spots: defects
(here they are barriers of Vb=0.3eV)
Blue region: channel
52
Conclusions
 Nanostructures offer the additional length scale degree of
freedom in design
 Thermoelectric properties can be largely improved.
 Very low thermal conductivities can be achieved
 Very high power factors can be achieved
 Advanced simulation techniques are required
 Atomistic models for electronic and phonon bandstructure
 Linearized BTE, Monte Carlo, NEGF methods for transport
 Future work
 Use all appropriate design guidelines to target TE
performance of nanocomposites (bandstructure, doping, κl)
 Collaborate with experimentalists and material scientists to
establish technologies
53